21 PRODUCT OF MEASURES 223 belongs to </(X), and we have (1 3.21 .3.1 ) J*A dv = A* ( f V, (with an abuse of notation analogous to those above). By (12.7.8) there exists an increasing sequence (hn) of functions belonging to JTR(X x Y) such that h = sup hn. For* each n, it follows from (13.21.1.3) n that the function /„(*) = \hn(x,y) d^(y) belongs to JfR(X); hence /= sup/n J n belongs to ./(X), and moreover we have/(x) = (*h(x, y) dn(y) for all x e X, by virtue of (13.5.2). Since v(hn) = A(/n) by definition, another application of (13.5.2) completes the proof. From now on, up to and including (13.21.16), we shall assume that the measures A e M(X) and \JL e M(Y) are positive, and we shall write v = A ® /i. (13.21.4) Ifh is any mapping ofX x Y into K, then (13.21.4.1) f */i dv £ A*( f *h(x, Let u e ,/(X x Y) be such that h g u. Then for each x 6 X we have Tsk /*H< /z(x, j) rf/4(.y) ^ n(x, y) d^(y\ and consequently = v*(u) by (13.21.3). Hence the inequality '(13.21.4.1) follows from the definition of v*(A) (13.15.5). We shall write Jf *h dX dfi or f f *A(Jc, y) dX(x) d/j,(y) instead of v*(A), and f *rfA(x) f */z(x, jj;) rf/i(^) in place of A*f |*/z(x, 7) <^(j)). Similarly, we shall use the notations 1 1 hd^d^ and j I h(x, y) dX(x) dfj,(y) for lower integrals. Thus the inequality (13.21.4.1) takes the form * Pd^x) (*h(x, (1 3.21 .4.2) x, y) <W(x) d^y) * d^x) h(x, y) dfi(y) with equality if h e */(X x Y), There is of course an analogous inequality (resp. equality) obtained by interchanging the roles of X and Y, and anuch that the oscillation of h in